No more dodging Algebra dilemma

For nearly two years, California’s unwieldy eighth grade math standards have lain untouched like an unexploded IED, a roadside bomb of the math wars. But with middle and high school math teachers clamoring for guidance and new assessments two-plus years away, the Legislature and State Board must soon answer the question, What about Algebra I in eighth grade?

Faced with political pressure from Gov. Schwarzenegger and bound by restrictions of the Legislature, the California State Academic Standards Commission and the State Board couldn’t resolve the issue in July and August 2010, when they adopted the Common Core standards in math and English language arts. A strong-willed minority of Schwarzenegger appointees to the Commission who had a hand in designing the 1997 state math standards – Ze’ev Wurman, a Palo Alto engineer, and Hoover Institution scholar Bill Evers – wanted to make Algebra I the default curriculum in eighth grade. The majority supported Common Core’s eighth grade standards, which introduce elements of algebra and geometry with the goal of sending students to high school better prepared for Algebra I and higher math.

So the Commission, whose job it was to advise the State Board, adopted essentially two courses worth of standards, the 28 Common Core eighth grade math standards and an Algebra I course with an intimidating 72 standards – an amalgam of a few of the old California Algebra I standards and Common Core high school algebra standards on top of Common Core eighth grade math. ***

The State Board, restricted by the Legislature to either adopt or reject – but not change – the package, adopted them intact on Aug. 1, 2010. That was the deadline for approval in order to get points for Race to the Top, which Schwarzenegger was pushing.

At that point, the Commission went out of business, leaving the State Board with no authority to modify the standards. Since then, eighth grade math has been a void. It’s not part of the Common Core interim materials adoption process, and there’s been confusion over how to create curriculum frameworks and teacher training for that grade.

A new Commission’s charge

Fast forward to this past Wednesday in the Senate Education Committee and the 7-2 passage of SB 1200, authored by Sen. Loni Hancock, an Oakland Democrat, on behalf of Superintendent of Public Instruction Tom Torlakson. It would establish an 11-member standards review commission charged with making recommendations to the State Board for modifying eighth grade math standards by July 2013.

The Legislature and Torlakson will name seven of the 11 members, with Gov. Brown naming the other four. The Commission and ultimately the State Board must decide whether eighth grade Common Core or a new Algebra I will be the default course ­– and how California will assess any standards that are outside of the Common Core. But it’s clear, from the language of the bill, what the Legislature’s intent would be: Common Core, not Algebra I, in eighth grade for most students.

“The rigor of the state Common Core standards is maintained so that all high school graduates are prepared for college and careers, as specified in the Common Core standards.”

All of the Common Core standards are adopted.

Modifications total no more than 15 percent of the already adopted state Common Core standards.(That percentage was the same limit imposed to qualify for Race to the Top.)

Retreat from universal Algebra in eighth grade

Wurman said Thursday that passage of the bill would confirm what he had predicted after the adoption of Common Core standards: Within a few years, there will be a sharp decline in the number of students taking Algebra I in eighth grade, leading to fewer students taking Advanced Placement Calculus in high school. Only students who are tutored or go to private schools that ignore Common Core will take Algebra in eighth grade, he said. “Private school kids will have calculus. Public school students will be less competitive for select private universities.”

California is one of the few states that adopted a policy of universal Algebra in eighth grade, and by some measures, it has been a marked success. Last year, two-thirds of eighth graders took either Algebra or Geometry – compared with only a third in 2003. Despite that doubling, the proportion of students who tested proficient rose from 39 percent in 2003 to 47 percent in 2011.

But consider the majority who aren’t proficient on the standardized test and even some who are, said Scott Farrand, a math professor at California State University, Sacramento, and leader of those on the Academic Standards Commission who favored Common Core math. “Tens of thousands of students now in Algebra I cannot add fractions,” he said. “The push for Algebra I is failing lots and lots of students.” Many of those forced to repeat Algebra I in ninth grade get frustrated and develop a dislike of math, he said. Common Core, with a more logical sequence and focus on understanding concepts starting in lower grades, will better prepare most students to succeed in Algebra and beyond, he said. “We want to build a system that allows them to move forward.” That’s why he disputes Wurman’s contention that fewer students, including minority students, will pursue majors in STEM – science, technology, engineering, and math – in college.

Can it be fixed?

Wurman and Farrand agree that the 72-standard Algebra I course that the Academic Standards Commission created is unmanageable, but they disagree as to how it came to be that way and whether it’s fixable..

Wurman and Evers argued for pushing down a number of Common Core standards to lower grades, from eighth grade to seventh and seventh to sixth, in order to prepare students for Algebra. But, with a handful of exceptions, the Commission refused, because members said, they didn’t want to tamper with Common Core’s order and sequence. (What to do with these non-conforming, acceleration standards will be the job of a separate group reporting to the State Board, the Instructional Quality Commission. It will create detailed grade by grade curriculum frameworks.)

By creating no on-ramp to Algebra, Wurman said, “we ended up with a fake algebra option that is infeasible.” At this point, the intellectually honest thing for the State Board to say is, “Our expectations of 8th graders have dropped. We screwed up and do not offer Algebra as an option.” The Algebra I course that the Academic Standards Commission designed would be taught in ninth grade.

Farrand said that the Algebra I course was voluminous because the Commission believed it had the authority only to add to Common Core standards, not eliminate them. A manageable Algebra I course for eighth grade can be created, he said, by pulling out some less related and duplicative standards, including probability and statistics. There should be an option for those students in a position to accelerate, he said. How it might be assessed is another issue for the new commission. Under the No Child Left Behind law, the federal government insisted on one test administered to all eighth graders. But that policy could change, because it’s not in anyone’s interest to discourage students from taking Algebra early.

Farrand and Wurman agree that the next Commission won’t be as contentious as the last one. For starters, neither Wurman nor Evers will be appointed. “I’m hoping the Commission will do something other than put on armor and fight,” Farrand said.

“No serious changes will be made to the standards. There won’t be anyone willing to go to the barricades” to defend rigorous California standards, Wurman said.

*** Here are the Common Core Standards as adopted in California. Start on page 34 for eighth grade. For the new Algebra I course, see page 36. Common Core high school standards are in yellow. Common Core eighth grade math standards are in green. California Algebra standards that have been included in the new Algebra I are in purple.

40 Comments

In some senses the bomb has already gone off. As a prior DAC member this topic has generated more questions for me than any other topic. For parents who haven’t been following common core the confusion around this topic has been a rude awakening. Rumors abound and school administrators don’t have a concrete response because of state level inaction.

“Tens of thousands of students now in Algebra I cannot add fractions,” he said. “The push for Algebra I is failing lots and lots of students.”

I dont understand that sentence. Adding fractions is learned in 3rd and sometimes even 2nd grade. Also, there are almost a half million kids at the 8th grade level. ‘Tens of thousands’ seems hardly a representative sample for the policy’s impact. Perhaps there is a better example?

Navigio: Scott Farrand was speaking as a college professor who speaks frequently with public school math teachers. Their biggest complaint is that they have to spend considerable time going over basics that they should have learned earlier, fractions being the biggest gap in their knowledge. He and others say Common Core spends more time focusing on concepts, fractions especially, so that students have a better understanding and grasp of math by the time they hit eighth grade. I’ll let him speak further, if he wishes.

Perhaps we should look to other countries when it comes to the teaching of math as opposed to when Algebra I is required. The stepping stones for Algebra can be introduced starting in earlier grades without impacting the teaching of numeracy.
In Alameda we have been working with Phil Gonsalves to provide professional development and coaching for elementary teachers. We have seen encouraging results with 97% of our students having taken Algebra I by 8th grade with proficiency levels above the state.
However one unintendend consequences is the difficulty of articulating math courses for high school students to fulfill graduation and college entrance requirements.

“Under the NCLB law, the federal government insisted on one test administered to all eighth graders.”

This statement is not true. North Carolina has a 2-test scheme [a full Algebra I test and an Algebra Lite test] that was approved by the feds in 2006. It has a shared scale of measurement [in technical terms, a vertical scale] linking the two tests to allow for a single set of cut scores and performance standards applied to both tests. CA can do the same for 8th grade math with the current Algebra I standards-based test for the 2/3rds of CA 8th graders now taking Algebra I and an Algebra Readiness standards-based test based on Algebra Readiness curriculum frameworks, instructional materials, and professional development approved by the State Board in 2007. The statement quoted above is continuation of a myth perpetuated by CDE staff when the Grade 8 Algebra issue came to a head during the summer of 2008.

The larger issue, however, is whether CA’s goal for Algebra I for 8th graders, established back in 1997, has been a success. It is hard to argue with the big picture facts generated by STAR trend data since the standards-based tests were installed in 2003 – In 2003, 1/3 of 8th graders took Algebra I and 39 percent scored proficient or above, and in 2011, more than 2/3 took Algebra I by 8th grade and about 50 percent scored proficient or above [the 2011 data includes 7th graders taking Algebra I in 2010]. These data document a stunning success story doubling the number and percentage of CA’s 8th graders taking Algebra I and significantly increasing both the number and percentage of those students scoring proficient or above on the Algebra I standards-based test. Yes, there have been some poor implementation practices in local districts and schools pushing 8th graders not yet ready for Algebra I into a full Algebra I course, but overall the Algebra I for 8th grade initiative in California has been a major success story. It needs another 10-15 years for full fruition.

To back away from the goal of Algebra I for 8th graders in California is very simply to dumb down the expectations that were established 15 years ago. Rather than back away from the goal, the action needed is a mid-course fine tuning of implementation practices.

Does our goal and our minimum have to be the same? Why can’t we have a *goal* of algebra in 8th grade and yet place kids where they need to be? If half are ready for algebra, that’s great! Let’s run with that… without bludgeoning the other half who are not.

What about the 53% who are still not scoring proficient after being compelled into theoretical math at such a young age (many of whom don’t have the foundation from earlier grades to be adequately prepared for this artificially high 8th grade standard)?

If one buys the logic of those favoring more rigorous standards, then perhaps the answer for this other half of struggling students is to raise 8th grade math standards to AlgII/Trig … or maybe go all the way to Calculus?

Or perhaps the better answer is to reconsider how we are teaching math, get away from arbitrary and decontextualized coursework, and provide more real-world applicable mathematics concepts to students (something that will stick with them for life, not just for the next fill-in-the-bubble regurgitation exam).

Oh, but many of those classes won’t prepare students for high school A-G coursework — the argument goes — and “A-G for all” is a preoccupation of the elites governing K-12 curriculum. So let’s just keep pounding on “those kids” who don’t grasp theoretical mathematics until they either learn to get along or drop-out (and one of the benefits of their early departure will be to help raise the test scores of schools … yippee!).

This isn’t a game … these are real kids with either bright and productive futures or dependency, despondency and prison. That’s the real forced tracking going on here.

John Good article. While it is true that a portion of students taking Algebra in the 8th Grade do not have the skills necessary to effectively compete, taking Algebra out of the 8th grade as called for by Common Core could have very serious negative consequeses for African American and Latino students. For years the dirty little secret of education was that we always said that these students “were not ready” for higher order math. Thus they never took Algebra or they took Algebra so late in secondary school they could not take the requisite higher math sequence and thus did not qualify for Universities. The fact is the the pressure to take Algebra for most students in the 8th Grade has had incredible success. 8 years ago fewer than 24% of African American students completed Algebra by the end of the 8th Grade. In 2011 58% took Algebra befor the end of the 8th Grade. The number scoring proficient or above increased from 1,678 students to 6,116. For Latino students the precentage taking increase from 22% to 62%. And the number scoring poficient or advanced grew from 10,178 to 58,946. So these students who were “not ready” actually were. To throw out this most secessful educational reform for African American and Latino students would be the biggest tragedy ever in California. Ask those professors and experts you talk to about this. Sure we should improve math achievement in K-7 and beyond but Algebra works better than any other state reform in history.

“What about the 53% who are still not scoring proficient after being compelled into theoretical math at such a young age (many of whom don’t have the foundation from earlier grades to be adequately prepared for this artificially high 8th grade standard)?”

I don’t believe anyone wants to “compel” unprepared students into Algebra — we want everyone to aspire to it, and be prepared for it. But if they are not, they should not be placed in Algebra. That is what the framework clearly states, that is what we’ve tried to do over the last 15 years. (And let’s ignore the 2008 momentary push to make Algebra mandatory, caused by a joint Federal and California stupidity; thankfully stopped by the court)

Further, Algebra I is not “artificially high” for 8th graders. Unless you think that American students are “artificially dumb” as compared to their Japanese, Korean, or Singaporean peers.

Finally, instead of using heated rhetoric such as “What about the 53% who are not scoring proficient [or above]” we should consider the following points.
(a) That the fraction of failure in Algebra I in grades 9 (77%), 10 (87%) or 11 (91%) is much higher than that of grade 7 (16%) and 8 (53%). Whatever else, it doesn’t seem to indicate that the early taking of Algebra is the problem.
(b) That many of other courses have similar “failure” (i.e., counting “basic” as “failure”) rates in high school. Geometry: 9 (51%), 10 ( 83%); Algebra 2: 10 (57%), 11 (84%).
(c) Should we rather keep 47% (using your numbers) in dumbed down courses in grade 8 instead of offering them what they are clearly capable of, and what their overseas peers are routinely taking?

I am baffled to read what Ze’ev Wurman just wrote – I worked as an educator in two CA districts where underprepared 8th grade students were forced into Algebra 1, including learning disabled students with IEPs, with the understanding in both districts that this was CA education policy. These students were miserable and many tried their utmost to make the teachers and other students miserable along with them. Every math teacher I know complains about compelling underprepared students to take Algebra 1 in 8th grade – including the high school teachers who get them the next year. Or two. Or three.
Is this disconnect between what he says should be happening and what is happening for real?
I think the CCSS math standards will likely better prepare all students, including non-white students, for learning algebra, and thus will be the better approach.
Ze’ev Wurman, in my 15 years as a teacher, I encountered a substantial percentage of 8th graders who were not ready for Algebra 1, and I do not think they are “artifically dumb.” I don’t know all the reasons why they weren’t ready, as they were often good in social studies, reading, art, etc. - I caution you to think twice about using an implication that students are stupid to make a point.

Ze’ev makes a critical point that I would like to expand on briefly. IMHO, this is one of the real flaws of our accountability system, and something anyone who discusses this issue really must understand. The failure rates in higher grades is a direct result of different kids taking the tests. This could not be more important to understand. It is just as EL proficiency rates in upper grades exclude the ‘successful’ ELs who have already ‘tested out’ of the program.

While I agree with that it is important to look at these passage rates by grades to have a better understanding of the pattern, the fact is that almost 60% of students take the Algebra I CST in 9th grade or later. And even around 5% of students take it in 7th grade, with an 83% proficiency rate I might add. In other words, at lower grades, its generally kids who are prepared for the test that take it, while later grade takers are those who either were not prepared earlier (or might never be), and their scores likely reflect that. If you look at EOC proficiency rates for Algebra I, they were at about 32% last year for the state overall, even with the much higher passage rates in lower grades. This same kind of pattern applies to most of the CST assessments that are not grade-restricted (not all, but most–science can show a slightly different pattern).

Furthermore, if you look at schools independently, you will find some wont encourage taking the Algebra I CST in middle school (I believe some might not even offer it!), while others will already have the majority of their students taking the Geometry CST in middle school. Of course, district-provided aggregated “math” proficiency rates by school might not bother trying to make that point, even though it often makes direct comparisons between schools pretty much meaningless.

Some time ago, I argued in this forum that using the CST as an indicator of “proficiency” is dangerous since, at least for all ELA and math lower than 7th grade, the distribution of scores is a bell curve with the average set as the proficient cut off point. By this measure, I wrote, half of CST takers are not proficient and, as interpreted by, for example, the Los Angeles Times, they are not at grade level. That did not go well with some who elected to argue that the average was increasing (2% a year, a rate that would require 20 years to reach “proficiency for all” if it is sustainable).

Anyway, given that the arguments in “favor” of Algebra I rest on this assumption on proficiency, I dare say that we, collectively, are asking for trouble if we wish to use the CST results as “the standard” method of assessing “mastery.”

But let’s take the CST score distributions at face value. What the test results are telling us is that there is a subset of students who can take Algebra I at 8th grade (and even 7th) and score higher than the “proficient cut-off.” But this doesn’t mean that all other students will be able to do the same. A perfect example is offered by the scores obtained by students at the Los Angeles Center for Enriched Studies, which is a 6-12 school in LAUSD and is, by now, filled with highly gifted/achieving kids.

There, a significant portion of 7th graders are encouraged to take Algebra I. 37% of them took Algebra I in 2007, 48% in 2008, 47% in 2009, and 31% in 2010. In 2007, 81% were proficient and above, while 90% were in 2008, 88% in 2009 and 96% in 2010. These are remarkable scores.

In contrast, 80% of 8th graders took Algebra I in 2007, 63% in 2008, 53% in 2009, and 56% in 2010. Of these, 68% scored below proficient in 2007, 62% in 2008, 60% in 2009, and 51% in 2010. These are also remarkable scores, but they are going in the opposite direction.

As a person trained to analyze data and question patterns, this tells me that kids (and adults, for that matter) are not all the same when it comes to “mastering” algebraic concepts. It is not that they are dumb (the scores for these same population in the ELA CST are in the 70 to 80% proficient and above). It is just that either 1) they can’t “do math” (just like I can’t “do music”), or 2) they have not received training that allows them to “do math.” My understanding is that LACES was starting to make inroads into that, but, unfortunately, the Title I funds they were using for that program have been cut off completely by the District starting 2012-13.

In summary, I believe we have to change the way we teach math and the way we measure math achievement. Else, we will be repeating the same mistakes again and again.

Navigio makes an excellent point that when we look at passing rates at different grades, we must consider the fact that the better-prepared students are those who tend to take the subject earlier, and hence self-select out of the group of those who take the test later. There is no easy way to correct for it (more on that below) and hence most arguments based on passing rates are quite weak. My response above was not intended to make a strong case, but rather was intended to respond to the even weaker case Fred was making when he argued that “53% is terrible.” All I was trying to show is that it is questionable that earlier Algebra taking is the reason behind failure rates as opposed, for example, to the impact of lowering assignment selectivity by preparedness. And we all know that misguided course assignment policies still exist in some districts, as many commentators have observed.

What we can observe is the fact that California did change the K-7 preparation since 1997, to prepare more students for Algebra in grade 8. And that this policy seems to have worked to a large degree, as the enrollment quadrupled (17% to 68% between 1998 and 2011 taking A1 by grade 8) simultaneously with passing rates monotonously increasing. John Mockler also pointed to its exemplary effect on minority students, who increased their participation and success rates much faster than the cohort averages.

The opponents point out that the changes are not perfect. In that they are right, except that they neglect to mention that no reform is ever perfect. The only way to get 100% success rate is to lower the expectations so low as to make them meaningless. What the opponents try to show, but can’t, is that the earlier Algebra — with appropriate K-7 preparation — causes harm in addition to its clearly proven benefits.

On a separate note, there is a new and interesting report that tries to figure out if taking early Algebra is actually harmful to at least some students. It uses a particular policy change in NC around 2001/2 that pushed a lot of unprepared students into early Algebra as a “natural experiment,” and it tries to tease out the lasting impact of this early assignment on unprepared students. While it is a fascinating and sophisticated bit of research, we should remember that it describes a situation where the K-7 content was not adjusted to support Algebra in grade 8. This is very different from what we’ve done in California since 1997 where much of the focus in K-7 was explicitly to make students ready for algebra in 8.

But all this may be going away as the Common Core explicitly does not attempt to prepare students for Algebra in grade 8. And once we completely remove the expectation of “Algebra in 8″ from California public schools, we are back to pre-1997 with only the privileged handful being able to take it.

I think the point about the adjustments in the K-7 curriculum is spot on. I’ve watched my daughter progress and the current math program in her school is definitely far more algebra-friendly than what I was taught with. (My mother was a math teacher, and I well remember her laments about how unnecessarily terrible the old curricula were at algebra preparation; many of the changes she advocated for are in the work my daughter brings home.) The last curriculum change in our district was I think 4 years ago and that also was a significant realignment. I always suspect those realignments pull down comprehension for the kids in the upper grades for a year or two, but for the benefit of all in the long run.

I like the way the new curriculum slides in algebraic concepts in a totally unobtrusive way starting in kindergarten. Elements as simple as having the answer sometimes on the left of the equal sign make a difference in how brains get wired.

All that to say that when looking at success rates, it’s not clear that what we’re doing today is reflected in 8th grade scores from even 5 years ago.

Manuel, I agree, especially on the way we measure. I would also (re-) point out that the 83% proficiency rate for 7th graders on the Algebra I CST is state-wide! So I dont see the scores you mention from a school full of high achievers to be necessarily out of the ordinary (statistically speaking of course ).

Also, I would tend to agree with the 2nd option in your analysis: ‘they have not received training that allows them to “do math.” ‘

Mike McMahon’s Alameda-boosting comment is nonsense. At Alameda Unified, a substantial percentage of students who took Algebra I in the 8th grade re-take it in 9th grade. Courtesy of Mr. Gonsalves, a coaching consortium from West Contra Costa County, and friendlly advice from the Alameda County Office, Alameda Unified follows a pacing guide that runs at breakneck speed. Everything is “covered” before the CST is administered. Harry Wong refers to the ‘myth of coverage’, just as scholars like Marston caution against superficial coverage of large numbers of standards. If you pull the pacing guide from the ACOE Web site, you’ll see that the entire last month of the Algebra I program is unspecified! If you pull the CST results from the CDE Web site, you’ll see that 62% of the 8th-grade Algebra I students are not proficient. Examining the “students tested” cell for each grade reveals the high repeat rate. This is hardly cause for celebration.

Fred Jones and PJ Hallam have it right about rushing students into Algebra I and using decontextualized curricula. Note that CPM is the only secondary math curriculum that will not have to be fundamentally rewritten to comply with the Common Core standards.

And nothing in the discussion of college readiness accounts for the fact that many students, while capable of attending college, choose not to put in sufficient effort.

Navigio, I was not interested on the breakdown by grade level when I looked at the STAR site. I was after what happens at this particular school which is supposed to be one of the top magnets in LAUSD.

But now that you point it out, it seems to me that it is obvious: only math high achievers are taking the Algebra I CST in seventh grade. No average student is going to do that and the fact that in 2011 only 7.7% of them did it confirms they are not average. (More damning to the notion that “everyone must take Algebra I in 8th grade” is the fact that 31.8% of 8th graders were deemed by their schools not to be ready for Algebra and took General Mathematics to be better prepared. My opinion is that this is not bad in itself, it just points out the need for better teaching as well as allowing for possible developmental differences.)

As for your agreeing with me that they haven’t received training, that’s another major problem: we expect ALL students to be doing relatively complicated math yet don’t want to spend money or create guidelines to ensure that something as simple as fractions and multiplication tables are thoroughly understood by 7th grade. I hear from the math teachers at my kid’s high school that many students don’t know their addition tables, much less their multiplication tables. I guess it is overreliance on calculators, but this points to the need for policy-makers to consult with those in the front lines of this “problem:” the teachers.

I teach algebra 1 and AP Calculus AB leveraging the knowledge I acquired while obtaining a bachelors in electrical engineering, an MBA in finance, and most recently, an MAEd, from a credential program committed to social justice, as well as my quarter-century experience working in high tech.

Fortunately, I recently discovered my true passion in life: teaching students so they might make the most of themselves in life as capable, competent citizens, full of self-esteem and self-confidence tempered by some humility and empathy for those who have less, and eager to serve their nation and this world in their life, whatever career path they choose. While the subject I teach is mathematics, it is simply the conduit for my wish to impart to students, of all socioeconomic backgrounds: knowledge, the ability to think independently and in groups, the confidence to make mistakes and fail while pulling oneself up so as to never give up entirely, the wisdom to seek help, and the desire to do your best. I especially wish to help underserved students by instilling in them the belief that they can overcome self-limiting thoughts and behaviors.

Having presented my bona fides, and expressed my intense desire to help all students, I must say that placing students deficient in prerequisite skills into algebra 1 (or any subject for that matter), whether in middle school or high school, creates a lose-lose-lose situation for the student, teacher, and society. Much of what I read on this site regarding student achievement, teacher effectiveness, closing the achievement gap, and etcetera seems to place undue emphasis on standards as if they are the primary reason for the success or failure of our educational system. Nothing could be further from the truth. They are a necessary but not sufficient condition for success. Many other factors impact the true success of this complex system spanning in-school and out-of-school environments including but not limited to robust standards, flexible curricula and pedagogy, dynamic and committed teachers, effective operational support systems, involved and supportive parents, and most importantly, students that recognize the significance and importance of learning so they serve as active, engaged participants with parents, teachers, administrators, and all others dedicated to providing world class education to our citizens.

Specifically, for algebra 1, the breadth of its twenty-five plus standards limits the opportunity to investigate all but a few in any reasonable depth, or at a pace reflective of the diverse range in student learning. Hopefully, with the continued advance of adaptive, online learning software, students who wish to learn may one day have the time and support they need to succeed, as opposed to expecting that through some miracle a teacher will rectify the myriad mathematical misunderstandings present in today’s algebra 1 classes. I consider myself very knowledgeable, highly capable, and extremely dedicated to helping all students. Yet, I realize an insurmountable task when I see one, especially when considering the entire system from end to end. Blame it on the engineer in me. While my heart aches to help all students overcome their challenges, it is clear that burn out is the surest outcome for the advocated approaches in today’s mainstreamed, heterogeneous classrooms where language, cognitive limitations, and other factors conspire to limit sustainable and extensible success. Students must be matched with material that aligns with their Zone of Proximal Development (Vygotsky, 1978) for true progress. Unfortunately, the range of ZPD within a classroom is daunting to say the least. While differentiated instruction is espoused to address that issue, it is unrealistic to expect teachers within their first five, or even longer, years to master it as there is a significant overhead associated with mass customized instruction. Whether anyone can do so is even debatable.

The following excerpts from my blog post: VAM BAM: Our National Obsession with Measuring Teacher Effectiveness expand on these points.

As an example, requiring every student to take algebra in eighth grade to ensure they have the greatest chance of attending college, while egalitarian on its face, results in the unintended labeling of a hundred thousand or so students as failures, and if using VAM, thousands of teachers as ineffective. The reasons so many students fail to learn algebra range from a student being unready developmentally, as recent advances in pre-frontal cortex research on cognition has revealed, to the concomitant challenges of poverty where a dearth of support resources and an overabundance of destructive environmental issues work to impede student learning.

As a first year algebra 1 teacher, I struggle with balancing the nearly unreasonable expectations placed on my students and myself alike, as embodied in content standards; federal, state, and local mandates; and the California Standards for the Teaching Profession (CSTPs). Many of these expectations cannot be successfully fulfilled in any sustainable basis given the current structure, roles, processes, and systems in public education, as compounded by the paucity of funding as measured per student.

As I shared with an acquaintance yesterday, if private industry had to perform under the same conditions, constraints, imperatives, policies, and expectations as teachers or administrators, we would not have profitable industries, much less profitable companies. Imagine depression era unemployment and Soviet-style stores as an outcome, for no one would deliver products to market that met customer expectations whether they be price, availability, or functionality; hence, jobs would be scarce, as would capital. Such is the state of public education today with little hope for true, mass scale improvement on the horizon. I say this since most education reformers believe that privatizing education is the way to go. However, the profit motive conflicts with the educate all ethos of public education. Anyone who says otherwise needs to spend more than a fleeting moment in public school classrooms.

I will restate my point, for clarity. There are approximately 200,000 8th graders each year who are not prepared to succeed in an algebra course. 8th graders who are ready for algebra should have access to a good algebra course and they should take that course. Students who are not ready for algebra in the 8th grade should not take algebra. Students who take algebra when they are unprepared for it tend to do very poorly in mathematics thereafter, so the stakes are much higher than a lost year of mathematics that can be retaken the next year.
Students who cannot reliably work with fractions are not ready for algebra. There is plenty of research to support this. I mentioned the inability to add fractions because it is a clear indicator that a student should not be in an algebra class. Ask some 8th grade algebra teachers whether they reteach fractions and then you might understand that adding fractions is not “learned in the 3rd grade.”
One reason for the abundance of students in 8th grade algebra classes who aren’t prepared to succeed in those classes has been the state’s push for 8th grade algebra for all. I would prefer a system that does a better job of preparing students for success in algebra and other coursework leading to college and career, and that does not place students into courses that they are not ready for, and that provides challenging courses for all students, throughout their school years. I don’t think that is controversial. How to do that is very controversial.

My previous comment was intended as a reply to the question from Navigio about the meaning of my statement about students who cannot add fractions who are in algebra classes. I didn’t realize that new comments all go to the end of the line.

This is a reply to the question from John Mockler about the success of the push for 8th grade algebra for all among minority populations.

To be sure, there are many more 8th grade students from low SES neighborhoods who now have access to algebra, and I believe that this is an intended benefit to the state board’s push for algebra for all 8th graders. I am not arguing that this policy is without benefit. It is also true that the students who are most likely to be misplaced in an 8th grade algebra class are African American and Latino students (see the EdSource study), so the policy is not without its faults.

I have not heard anyone suggest that 8th graders should be unable to take algebra. If prepared for it, they should be able to take it. Stepping back from making algebra THE 8th grade math course does not mean it cannot be an option, even an option that is taken by a majority of students. Two important questions are: how to provide a better preparation for students, so that more are ready to succeed in algebra and other courses, and how to ensure that all students have the option of challenging curriculum, regardless of the neighborhood they live in. I believe that the Common Core Standards will be a significant next step in answer to the first question. The second question is a policy question that I hope is being given serious thought. One suggestion I have heard is that in the next generation of accountability systems, rather than dinging schools for every student not taking algebra, they would be given credit for students who are successful in their algebra class. By not giving credit for students who do not succeed in algebra, this might avoid the pushing of students into courses in which they are unlikely to succeed.

@ Ze’ev, through conversations with experts from Silicon Valley Math, and see also http://www.cpm.org/teachers/CCSS_Practices.htm . From what I know of your work, you would probably argue that content is king, but I believe that process matters too. As a teacher, I believe that the “Standards for Mathematical Practice” are the exciting part of the Common Core. Unlike other publishers’ offerings, the CPM books (I referred to the entire secondary math curriculum, not to any single course) already fulfills them.

This is a reply to the question from el about where less time would be spent if there is a greater focus on fraction concepts.

In the Common Core math standards, there will be some changes to the order in which instruction occurs. For example, rather than teaching integer arithmetic (dealing with negative numbers) at the same time as students are learning to work with fractions, fraction instruction will be largely completed before negative numbers appear. Similarly, probability and statistics appear after fraction instruction in the Common Core. I believe that this arrangement was based on the practices of more successful countries.

The most significant improvement in fraction instruction that will come with the Common Core is the more detailed mathematical progression through the topics. The savings in time that I anticipate from this is in the time spent reteaching. Ask a math teacher sometime how much of their time is spent reteaching topics from earlier grades. Any improvement in this situation will be significant.

No matter the order or the textbook content, until elementary and middle school teachers are competent in mathematics, understand the developmental issues that require differentiated multiple approaches to teaching, and are organized within the school to catch areas of weakness there won’t be change for some students. Content and process cannot be separated.
I have watched students in a school cluster that is exceptional, and a good proportion of those students not only master Algebra in 7th, but Geometry in 8th. What the State should be doing is taking those teachers and students and learning what went right – then build in those strengths to schools where there are problems. Thank you to Ze’ev and others who worked to bring in those changes otherwise my oldest son and his cohort would not have experienced those opportunities to excel in the public system.
In all honesty, if the math teacher my oldest son had in 7th and 8th grade was filmed teaching, and her comments on typical errors and where they originate were noted, there would be material there to help others.
In the meantime – I still can’t get my younger kids’ school to stop using a 1995, non-aligned textbook for 6th grade, and student teachers in 4th and 5th are primarily teaching/running groups at the same time as fraction teaching … it’s not a good scenario. I cannot express how badly fractions are taught in some schools – and without understanding multiplication, number relationships, it all gets worse.

Using the term “prepared” is incorrect. The term should be ability. Most kids, in elementary school receive the same level of math instruction. When they reach middle school they are then assessed to ascertain their ability. My children, both of which are now in college (and were not required to take any remediation courses), both took algebra in middle school. The eldest also took geometry. The problem with taking geometry in middle school is that it forces a student into levels of math they may not have an interest in pursuing. Although some may argue that society gains as a result of these kids taking higher math and continuing on to university to major in a math related field, the reality is it is a burden on those not so inclined.
It is possible that being forced into higher math can negatively affect admissions and financial aid, if the student does not do well in theoretical mathematics.

One of the challenges of reviewing Math proficiency data is test taking for each grade level after 6th grade is hard to aggregate. In Alameda, students can begin taking Algebra I in 7th grade. After numerous refinements, school sites are getting better at identifying students who can take Algebra I in 7th grade. Therefore to truly evaluate the effectiveness of the Math coaching providing you need to evaluate the numerous cohorts.
Overall, I believe the Math coaching provided in Alameda elementary schools that started in 2006 is working and improving student outcomes in Math.

Regarding various comments about fractions, I wrote about a brief study I conducted with my three algebra 1 classes late last semester in the following post: Problems with Prerequisites: Part 1. It focused on the troubles students have with various prerequisite skills needed for success in algebra 1, to include fractions / rational numbers. Here’s a snippet from that post.

“As the data in the table show, students did not fare very well on much of the assessment. Even a class average of 80%, which might be considered impressive by some, signaled challenges in students’ fundamental mathematical proficiency as the complexity of the operations and numbers in each problem merited a higher score. In contrast to the semi-success with integer operations, student performance with fractions indicated significant struggles. Furthermore, and somewhat surprisingly, the simple presence of the variable “x” with the same integers and mathematical operations led to scores falling considerably, as shown in the following chart. Performance declined less from rational numbers to rational expressions since those students who struggled with fractions were already removed from consideration, which speaks volumes in itself.”

I offer no eurekas as I am so new to this challenge. However, it is abundantly clear to this former systems engineer, and business process analyst, that the one-size shoe fits all approach used at present is failing miserably. I have many ideas that are percolating away in my free time from midnight to 6AM but they are too ill-formed to share at this point.

Hi Scott.. sorry for the delay… if you base the assessment on CST proficiency then there are half a million kids every year that are never ready for Algebra 1, regardless of grade level. I obviously agree that there are differences in readiness, but I kind of doubt the majority that never make it to proficiency have somehow been ‘hindered’ by a curriculum that teaches fractions in 3rd grade (I should have said ‘taught’ instead of ‘learned’, since there is obviously a difference when looked at in hindsight).

I also am not convinced that its fair to try to determine ‘when’ to teach certain subjects based on whether remediation is needed later for those subjects. If so, does the answer have to be that it was given too soon? And if so, how does that jibe with the minority who succeed in algebra in 7th grade already and the half who succeed in 8th grade? Not to mention all the other remediation that happens at all points in secondary and post-secondary education (and by some accounts, thats been happening forever).

My personal take on the subject of remediation is that it is less a function of when the concept is introduced and more a function of whether those concepts are continually used in subsequent years. I’d bet that most people who need remediation in fractions when they get to algebra actually knew fractions at one point, but forgot them (and I think there is a reason for that). Anecdotally, in a 3rd grade class I am familiar with in a school with a high poverty and predominantly minority population, less than 10 percent of the kids cannot add fractions or dont know their multiplication tables to 12. Its clear they’ve ‘learned’ that. Statistically speaking, the majority of those kids probably wont be able to do either by the time they reach high school. If thats true, is it fair to say they never learned that?

And although you were not specific, I expect that the reason what you were mentioning might be controversial is that it would tend to encourage a process in which what manifests is a differentiation of expectations (not of support), and based on ‘criteria’ that is utterly irrelevant to such determinations. Personally, I think we have too much of that already, and I might even argue its part of the reason why we see so much variability in ‘ability’ by the time we are assessing for algebra readiness. IMHO anyway..

Regarding CPM, you write that “Unlike other publishers’ offerings, the CPM books (I referred to the entire secondary math curriculum, not to any single course) already fulfills them [process standards].”

Ay, there’s the rub. The eight “process standards” take 3 pages in CCSSM; the content standards take 75 pages. Claiming alignment because of those three pages is next to meaningless.

And let’s be a bit specific. These eight process standards are so generic and so mathematically obvious that EVERY TEXTBOOK AROUND can rightfully claim it is aligned with them. Here they are, for the uninitiated:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision.
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.

I doubt that there is a single mathematics book in this universe that, were you to ask its author or publisher, would not swear that his book addresses, and is guided by, those “process standards.” For any grade. So all publishers already claim (quite properly!) that their books are all aligned with the Common Core’s process standards. It is the trifling problem with the other 75 pages of the content that gives them some grief (smile).

To fix the problem of students failing Algebra is to do what Texas and other surrounding states are doing and have been doing for the last 20 years. They don’t take algebra until they successfully complete pre-algebra. I graduated high school with many guys who took pre-algebra for 6 years. They didn’t or couldn’t successfully pass pre-algebra so why challenge them to a much harder course. Mandating that all students take such rigorous course is not the answer to student success that many of us educators in CA may think.

I teach 7th grade regular math (using a Pre-Algebra textbook,) and this is why I think kids struggle with fractions and factoring:
Today fewer of my students have their multiplication facts memorized than when I taught nine years ago–even some of the good students. As a result, they struggle with fractions.
If one knows the multiples of a number by heart, it’s easy to:

find a common denominator,
cross cancel when multiplying
simplify
convert an improper fraction to a mixed number

I wonder if the focus on standardized testing has pushed elementary teachers to spend less time on helping students memorize the facts and more on learning the many procedures the students will be tested on.Implication for Algebra:
Not only do these students have difficulty with fractions, they also struggle with factoring, because they don’t automatically know any factors of a number. To them 18 is just that–18. Their brains do not say 3 x 6 or 2 x 9. If you ask them what 2 x 9 is, they will pause (to count by twos on their fingers) and then give you the answer in an unsure voice. “18?” they’ll say, or sometimes “16?”Conclusion: I hope elementary teachers will encourage students to memorize their multiplication facts, even though this takes time away from teaching to tested standards; then these students can have great success in Algebra, when they get there. They’ll see 16x + 28 and think, “4 x 4 is 16 and 4 x 7 is 28, so I can factor out a 4!” And they’ll find that math is really fun. I’m guessing that all of us here love math, and I think it’s partly because some wonderful people helped us memorize our facts!

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